## [PDS] Variance of Increments of Brownian Motion

**Proof that Variance of Increments of Brownian Motion is **

Consider the Brownian Motion at times and . We can interpret the value of the Brownian Motion at these times as limits of a discrete time Random Walk at and with payoff scaling as and respectively, i.e.:

where the Variance of the Brownian Motion till times and is respectively given as:

The increment in the Brownian Motion over time is a difference of the two limits, i.e.:

We are interested in the Variance of the increment of the Brownian Motion:

where in the above steps we have used the following:

**NB:** in Step 5 holds because, by construction, the Random Walk from time to time is independent of the Random Walk till time .

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